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Solomon's seal knot is the prime (5,2)-torus knot with braid word . It is also known as the cinquefoil knot (a name derived from certain herbs and shrubs of the rose family which have five-lobed leaves and five-petaled flowers) or the double overhand knot. It has Arf invariant 1 and is not amphichiral, although it is invertible.
The knot group of Solomon's seal knot is
(1)
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(Livingston 1993, p. 127).
The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F of the Solomon's seal knot are
(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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Surprisingly, the knot 10-132 shares the same Alexander polynomial and Jones polynomial with the Solomon's seal knot. However, no knots on 10 or fewer crossings share the same BLM/Ho polynomial with it.